Singularities of local models

We construct local models of Shimura varieties and investigate their singularities, with special emphasis on wildly ramified cases. More precisely, with the exception of odd unitary groups in residue characteristic $2$ we construct local models, show reducedness of their special fiber, Cohen$-$Macaulayness and in equi-characteristic also (pseudo-)rationality. In mixed characteristic we conjecture their (pseudo-)rationality. This is based on the construction of parahoric group schemes over two-dimensional bases for wildly ramified groups and an analysis of singularities of the attached Schubert varieties in positive characteristic using perfect geometry.